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Abstract

Reflection, diffraction and transmission of optical waves at the interface between a photonic crystal and the surrounding air can be described by propagating and evanescent Bloch modes. We have found such modes for one of the canonical two-dimensional photonic crystals, identical circular cylinders in a square pattern. We present computed out-of-plane band diagrams for propagating as well as evanescent modes, obtained with a numerical method based on Fourier-Bessel expansions. For a given frequency, all the modes are evanescent, except for a few low-order propagating modes. We find that most of the evanescent modes have a purely imaginary z-component of the Bloch wave vector, but many of the modes have a complex z-component.

Figures (10)

Fig. 1. Plane waves in three dimensions in air hitting the surface of a 2D PC. We would like to calculate the waves that are reflected and diffracted from the interface between air above the xy-plane and the PC. To do so, modes for a 2D PC extruding infinitely in the z-direction are needed.

Fig. 3. Point matching for 12 sampling points around the unit cell of the PC, for the z components of the E and H fields (a) and for the transversal components (b). The small arrows in (b) indicate which transversal component that is matched.

Fig. 5. Automatically computed band diagram (black lines) for a low-contrast structure, ε1 = 4.0 and ε2 = 1.7. The unit cell geometry is as in Fig. 4. Green lines show plane waves for εavg = 2.0 (the average dielectric constant for the low-contrast structure). Figure (b) shows a magnification of the marked area in Fig. (a) where two of the bands form a complex-conjugated pair of k2z (marked with dotted lines).

Fig. 6. (a) The largest value of Im k2z in the bandgap as a function of the permittivity contrast Δ. (b) The upper and lower boundaries (of k20) of the bandgap and the bandgap size, as a function of the permittivity contrast.